I had something similar happen a while ago. In my case the OAuth login window wasn’t completing properly, so Action never received the confirmation from Twitch.
A couple things you could try:
Log out of Twitch in your browser, then try connecting again from Action.
Temporarily disable browser extensions (especially ad blockers) if the login window opens through your default browser.
Check in your Twitch settings → Connections and remove any old Mirillis/Action authorization, then reconnect.
If it still doesn’t work, using the manual stream key is a good workaround and usually works without issues.